Mathematics in Art and Architecture “The universe is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.”~ Galileo Galilei Mathematics and Art may at first seem to have nothing in common at all. There is actually a great deal of math involved in art, including basic things like lines, measurements, and angles. Often, people who enjoy math tend to look for mathematics in art. These people want to see the lines of perspective, the patterns and angles. This is why mathematicians like artists like M.C. Escher so much. Ancient Civilizations knew about and used the golden ratio, it was thought of as a ratio that is very pleasing to the eye. They used the golden ratio in great pieces of architecture. The Great Pyramid of Giza The Great Pyramid of Giza (also known as The Pyramid of Khufu) was completed around 2560 B.C., it is the oldest of the Seven Wonders of the Ancient World, and the only one to remain mostly intact. There is still some discussion as to if the Great Pyramid of Giza was built based on the golden ratio. The rough inner part of the pyramid remains, but the once flat outer layer is completely gone, which makes it difficult to know for sure. There is convincing evidence, however, that the design of the pyramid included the golden ratio. By applying the Pythagorean equation (a² + b² = c²) to phi (1.618… + 1 = 2.618) we can make a right triangle, of sides a, b and c, or in this case a golden triangle of sides √Φ, 1 and Φ. This forms a pyramid with a height of the square root of Phi, 1.272., and a base width of 2 (two triangles above placed back-to-back). The ratio of the height to the base is 0.636. "According to Wikipedia, the Great Pyramid has a bas... ... middle of paper ... ...believed that the essence of the physical world and the whole universe was all related in some way to mathematics. Albrecht Durer Albrecht Durer (1471–1528) was a German Renaissance printmaker who made important contributions to polyhedral literature in his book, Underweysung der Messung (Education on Measurement) (1525), meant to teach the subjects of linear perspective, regular polygons, Platonic solids, and geometry in architecture. Durer is also the first to write down the concept of polyhedral nets, polyhedra unfolded to lie flat for printing. Durer's famous engraving, “Melencolia I” depicts an exasperated thinker sitting by what is best elucidated as a “truncated rhomboid” or a “rhombohedron with 72-degree face angles, which has been truncated so it can be inscribed in a sphere”. It has been the subject of more interpretation than most any other print.
The Pyramids Of Giza were chosen as they are three extraordinary structures that give a rich insight into the context, culture, function, technology, power and experiential aspects of the time period and the buildings themselves. The three pyramids showcase the Egyptian’s advanced construction and design methods, their religious beliefs and practices, their rich and diverse culture, the power of the king, as well as the context that surrounded these magnificent structures.
Albrecht Durer completed the “Master Engravings” in the years 1513 and 1514. With these three engravings (Knight, Death, and Devil, St. Jerome in His Study, and Melencolia I) he reached the high point of his artistic expression and concentration. each print represents a different philosophical perspective on the “worlds” respectively of action, spirit, and intellect. Although Durer himself evidently did not think of the three as a set, He sometimes sold or gave St. Jerome and Melencolia I as a pair.
Though most works of art have some underlying, deeper meaning attached to them, our first impression of their significance comes through our initial visual interpretation. When we first view a painting or a statue or other piece of art, we notice first the visual details – its size, its medium, its color, and its condition, for example – before we begin to ponder its greater significance. Indeed, these visual clues are just as important as any other interpretation or meaning of a work, for they allow us to understand just what that deeper meaning is. The expression on a statue’s face tells us the emotion and message that the artist is trying to convey. Its color, too, can provide clues: darker or lighter colors can play a role in how we judge a piece of art. The type of lines used in a piece can send different messages. A sculpture, for example, may have been carved with hard, rough lines or it may have been carved with smoother, more flowing lines that portray a kind of gentleness.
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that...
Schattschneider, Doris. “The Fascination of Tiling.” The Visual Mind: Art and Mathematics. Ed. Michele Emmer. Cambridge: MIT Press. 157-164.
It required many steps. The first step is putting large stone blocks to the height they required. As the pyramid grew taller, the ramp had to be extended in length, and its base was widened, or else it would collapse. It is likely that for the construction of each pyramid, several ramps were probably used. The second step is the internal construction of Egyptian pyramids which consisted of a series of buttress walls surrounding a central core. The walls decrease in height from the center outwards; to be more clear, the core of the true pyramid is essentially a step pyramid. The internal arrangement added stability to the structure. Packing blocks filled the "steps" formed by the faces of the outermost buttress walls and casting block, often made of limestone, completed the structure of the true pyramid.
Leonardo Da Vinci is one of the few artists and mathematicians who used the Golden Ratio frequently. In the Renaissance, the Golden Ratio was often used to create balance and beauty in statues and and paintings. Da Vinci, however, called it the “Golden Section”. He used it in famous
While studying the golden mean it becomes evident just how relevant this number is in the world. Many architects and artists have used this ratio as a scale and proportion sequence. The sequence is also relevant in music, nature and even the human body. Ancient mathematicians were so fascinated in the ratio because of its frequency in geometry. The first person to provide a written definition was Euclid. He stated “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less” this has been studied thoroughly by many mathematicians but the most relevant was the studies of Leonardo Fibonacci. Fibonacci is famous for the work he put in to come up with the Fibonacci sequence.
is convergent and ends up converging to φ, the golden ratio [2]. This curious quantity is just a ratio, so what makes it so special? Why is it so mystifying? While the other major constant in mathematics, pi, is a ratio between a circle's circumference and its diameter, phi (φ) considers a rectangle with height, h, and width, w, and forms the following ratio:
After the death of his mother in 1514, Dürer created Melancolia I [figure 5], a copper engraving that is “widely considered the pinnacle of classical printmaking” (Chudnovsky, 2014). Moreover, the engraving is an excellent example of Dürer’s integration of mathematics into his art (Walton, 1994). Accordingly, Dürer was quite pleased with Melancolia I and he produced reproductions of it on the best paper available and gave them away as a form of self-promotion. The allegorical work is filled with mysterious symbols and Dürer never felt inclined to explain their meaning. As a result, Melancolia I has been the subject of extensive academic debate over the course of the past 500 years.
- [5] Ritter, M, The Great Pyramid of Khufu, Retrieved April 12, 2005, "The Great Pyramid of Khufu...is the largest pyramid in Egypt and was the tallest man-made structure in the World until 1888."
The construction of the Great Pyramid is still a mystery and marvel to this day, there are seven major feats that archeologists still don’t understand.
It is believed that the shape of the pyramid was an important religious statement. Some scholars believe that this is true while others still debate the possibilities. We can assume that the Egyptians were trying to symbolize the slanting rays of the sun. It is also believed that the sloping sides on the pyramid were intended to help the soul of the king climb to the sky and join the gods.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...